CN108764351B - Riemann manifold preservation kernel learning method and device based on geodesic distance - Google Patents

Riemann manifold preservation kernel learning method and device based on geodesic distance Download PDF

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CN108764351B
CN108764351B CN201810542508.2A CN201810542508A CN108764351B CN 108764351 B CN108764351 B CN 108764351B CN 201810542508 A CN201810542508 A CN 201810542508A CN 108764351 B CN108764351 B CN 108764351B
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CN108764351A (en
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牛菓
王修才
顾艳春
段志奎
陈建文
樊耘
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Foshan University
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Abstract

The invention discloses a Riemannian manifold maintenance core learning method and device based on geodesic distance, which are used for solving two problems of the core learning method of the Riemannian manifold data of a digital image: 1) the measure keeping problem is solved, the distance of the symmetrical positive definite matrix on the Riemannian manifold transformed to the Euclidean space is equal to the geodesic distance on the Riemannian manifold by a parameterized Markov distance learning mode, and the optimal solution of the Markov distance matrix is obtained according to a Bregman optimization algorithm; 2) the problem of the fixed core method. By way of kernel learning; according to the Bregman optimization algorithm, the kernel learning is carried out to obtain an optimal kernel matrix, so that the distance on the kernel space is consistent with the geodesic distance, and finally the problems of Riemann manifold measure keeping and single-kernel limitation are effectively solved.

Description

Riemann manifold preservation kernel learning method and device based on geodesic distance
Technical Field
The disclosure relates to the field of machine learning and image processing, in particular to a Riemannian manifold preservation kernel learning method and device based on geodesic distance.
Background
With the rapid development of multimedia technology and the rapid development of internet technology, the collection, storage, transmission and access of digital image information are explosively increased. Digital images are not only flooded on the internet, but also in the fields of civil, commercial, military, medical, biological and the like, resulting in a large amount of image information, including photographs taken in daily life, news pictures, medical pictures, biological pictures, remote sensing pictures and the like. In all of the above cases, the processing of images becomes more and more complicated. Because image data faced by image processing is large in quantity and high in dimensionality, if the image is processed by a traditional method, high time cost and undesirable recognition rate can be caused. For example, in a method commonly used for image processing, an image is converted into a row vector or a column vector, i.e., a row vector or a column vector is used to represent an image, and if the dimension of the image is relatively small or not too high, the amount of computation after conversion is not too large. However, with the rapid development of internet technology, images are clearer and have larger dimensions, which are no longer limited within hundreds of dimensions, but tens of thousands, hundreds of thousands of dimensions, or even larger. If the image is converted into a vector at this time and then applied to subsequent classification, recognition, and the like, the amount of calculation increases drastically, resulting in high time cost. Each image will have its own features, and in general, for an image, the common feature extraction methods are Covariance descriptors (document 1, Tuzel O., Portikli F., Meer F., Region Covariance: A Fast Descriptor for Detection and Classification, European Conference on Computer Vision (ECCV),2006), diffusion Tensor descriptors (document 2, Pennec X., Fillard P., Ayache N., A Riemannian Framework for software calculation, International Journal of Computer Vision (IJCV),2006), structure tensors (document 3, Goh A., Visal R., Cluster and dimension), texture descriptors (CVPR) and other features, and the like. The covariance matrix is used as a region descriptor, so that not only can a plurality of possibly related features in an image be fused, but also a single covariance matrix extracted from a region can be used for realizing the matching problem of the region under different visual angles and postures.
The measure learning and the nuclear learning are widely applied to machine learning. It is generally accepted that metric learning comes from work by Xing et al, 2002 (document 4, Xing e.p., Jordan m.i., Russell s., Distance measurement with application to calibration with side-formation. processing of the Advances in Neural Information processing Systems, 2002). Davis et al (literature 5, Davis j.v., Kulis b., Jani p., Information-to-the-electronic metric Learning, Proceedings of the International Conference on Machine Learning,2007) transformed the metric Learning problem into Bregman's problem and proposed Information theory metric Learning from an Information theory perspective. Thus under the distance function constraint, metric learning becomes a problem by minimizing the differential relative entropy between multivariate gaussians. Davis et al, by using Bregman's algorithm, formulated the metric learning problem as a special Bregman optimization problem, and finally learned the metrics between data by minimizing LogDet divergence. The existing measure learning method mostly depends on the Mahalanobis distance and is limited to low-dimensional data. Nuclear learning (document 6Shawe-Taylor J. and Cristianini N., Kernel methods for pattern analysis. Cambridge University Press,2004) is also widely used in machine learning. However, most of the kernel methods are direct-push methods, and thus cannot be effectively expanded into new data applications. However, the measure learning method is introduced into the nuclear learning, so that the existing nuclear method can be effectively improved. Some researchers improve the algorithm by combining kernel learning when metric learning is used as a linear transformation learning problem. The specific method is to express the learning problem as the problem of minimizing LogDet divergence under the condition of meeting a certain linearity condition. Generally, however, the linear constraints contain classification information and need to be performed in a fully supervised or semi-supervised situation.
Disclosure of Invention
The method comprises the steps of introducing the geodesic distance between any two points on the Riemannian manifold into measure learning to optimize an initial core matrix, further representing the problem of core learning as the problem of minimizing LogDet divergence under the condition of meeting a certain linear condition, and finally carrying out data classification according to the obtained core matrix. This is achieved byThe advantages of the sample are as follows: the proposed constraint condition is to keep the geodesic distance on the manifold, namely the distance between two points on the nuclear space equal to the geodesic distance, does not contain classification information, and is an unsupervised learning mode. At the same time, according to the obtained optimal Mahalanobis distance, the linear transformation K is brought into0＝XTA*X, mixing K with0And as an initial matrix, performing kernel learning, so that the finally obtained kernel matrix meets the condition that the distance between two points in a kernel space is equal to the geodesic distance, calculating a covariance matrix of the characteristics of each sample data according to a given data set, applying Riemann geodesic distance to measure learning, calculating the initial kernel matrix of the kernel learning problem according to the obtained optimal measure, and finally obtaining the optimal kernel matrix according to a Bregman iteration method.
In order to achieve the above object, the present disclosure provides a riemann manifold preserving kernel learning method and apparatus based on geodesic distance, wherein the method includes the following steps:
step 1, calculating a characteristic covariance matrix of each sample data according to a data sample set and a covariance descriptor;
step 2, calculating the Riemann geodesic distance according to the characteristic covariance matrix;
step 3, learning an initial kernel matrix defined by the Riemann geodesic distance by adopting a Bregman iterative algorithm to obtain an optimal kernel matrix;
and 4, classifying the data according to the obtained optimal kernel matrix.
Further, in step 1, the method for calculating the feature covariance matrix of each sample data includes:
calculating a characteristic covariance matrix of each sample by adopting a region covariance operator method, setting I as a sample picture of three-dimensional color intensity, and F as a characteristic image of W multiplied by H multiplied by d dimensions extracted from I: f (x, y) ═ Φ (I, x, y), if I is a three-dimensional picture, F is a feature image of a sub-image taken from I, the size of F is in dimensions W × H × d, x, y is within the size range W, H, d is the number of elements in F (x, y), the function Φ is one of the mappings of image intensity, color, gradient, filter response, general feature vector F (x, y):
Figure GDA0003150454250000031
(x, y) denotes a position of the pixel, R (x, y), G (x, y), B (x, y) denotes a color value of the pixel,
Figure GDA0003150454250000032
respectively, the norm of the first differential with respect to x, y,
Figure GDA0003150454250000033
the norm representing the second order differential with respect to x, y, respectively, for the texture recognition task, the feature vector F (x, y) is constructed:
Figure GDA0003150454250000034
where I (x, y) represents image intensity, and commonly used eigenvalues in constructing the eigenvectors are the image intensity values, the norm of the first and second differentials with respect to x, y, for a given rectangular region
Figure GDA0003150454250000036
{zkD-dimensional features in the region R, k is 1 … n, n is a positive integer, and the region R is represented by a covariance matrix of d × d of features in the region R:
Figure GDA0003150454250000035
u is the mean of the feature points, expressed for the (i, j) th element of the covariance matrix in the formula,
Figure GDA0003150454250000041
the spread average, the above equation may be changed,
Figure GDA0003150454250000042
calculating the sum of the features of each dimension
Figure GDA0003150454250000043
And the sum of the products of any two features, z (i) z (j), i 1 … n, j 1 … n, n being a positive integer, the regional covariance C can be obtainedR。
Let X be { X ═ X in a given dataset1…xnIn which xiRepresenting data samples, xiI is 1, …, n, n is a positive integer;
further, according to the characteristic covariance matrix, the Riemann geodesic distance is calculated. The characteristic covariance matrices obtained in step 1 are all symmetric positive definite matrices, and all symmetric matrices are a Riemann manifold. For any two symmetric positive definite matrixes Xi,XjPerforming a logarithmic operation to obtain Zi＝log(Xi),Zj＝log(Xj). After logarithmic calculation, Zi,ZjStill a matrix. Note that the logarithmic operation here is a riemann manifold special operator, which is data dependent. Further, handle Zi,ZjThe matrix is processed into a column vector
Figure GDA0003150454250000044
Newly obtained column vector
Figure GDA0003150454250000045
Can be regarded as a point in Euclidean space, and by using a parameterized Mahalanobis distance form, can obtain:
Figure GDA0003150454250000046
learning of the above matrix a can be solved with LogDet divergence. I.e. where A and A are for n matrices0A and A0The LogDet divergence is:
Dld(A,A0)＝tr(AA0 -1)-log det(AA0 -1)-n。
there is no feasible solution to prevent minimizing LogDet divergence, and a relaxation variable γ is added to the representation to ensure an optimal solution.In the invention, in order to achieve the purpose of keeping geodesic distance, the point of the transformed symmetric positive definite matrix is obtained after learning the mahalanobis distance on the Euclidean space
Figure GDA0003150454250000047
After adding the relaxation variable γ, the learning problem for the matrix a can be expressed as:
Figure GDA0003150454250000048
Figure GDA0003150454250000051
further, in step 2, the method for calculating the riemann geodesic distance according to the feature covariance matrix includes:
for any two symmetric positive definite matrixes Xi,XjPerforming a logarithmic operation to obtain Zi＝log(Xi),Zj＝log(Xj). After logarithmic calculation, Zi,ZjStill a matrix. Note that the logarithmic operation here is a riemann manifold special operator, which is data dependent. Further, handle Zi,ZjThe matrix is processed into a column vector
Figure GDA0003150454250000052
Newly obtained column vector
Figure GDA0003150454250000053
Can be regarded as a point in Euclidean space, and by using a parameterized Mahalanobis distance form, can obtain:
Figure GDA0003150454250000054
learning of the above matrix a can be solved with LogDet divergence. I.e. where A and A are for n matrices0A and A0The LogDet divergence is: dld(A,A0)＝tr(AA0 -1)-log det(AA0 -1) N, there is no feasible solution to prevent minimizing LogDet divergence, adding a relaxation variable γ to the representation to ensure an optimal solution. In the invention, in order to achieve the purpose of keeping geodesic distance, the point of the transformed symmetric positive definite matrix is obtained after learning the mahalanobis distance on the Euclidean space
Figure GDA0003150454250000055
After adding the relaxation variable γ, the learning problem for the matrix a can be expressed as:
Figure GDA0003150454250000056
where ζ is the variable ζijThe column vector of (a) is,
Figure GDA0003150454250000057
is the definition of the Mahalanobis distance between two points (vectors) in Euclidean space, dAThe geodesic distance between data on Riemannian manifold is Ma's distance, and can not be directly calculated by using Euclidean distance, so that here we firstly convert manifold data into Euclidean space, and use Ma's distance to be approximately equal to geodesic distance, i.e. use
Figure GDA0003150454250000058
To ensure, wherein dgesFor geodetic distance, the matrix a contains distance information for ranging.
Further, in step 3, the initial kernel matrix is defined according to the matrix A of the Mahalanobis distance
Figure GDA0003150454250000061
Further, in step 3, learning the kernel matrix by adopting a Bregman iterative algorithm to obtain an optimal kernel matrix K*Approximation K0The resulting kernel matrix satisfies that the distance in kernel space is equal to the geodesic distance in Riemann manifold, for n × n matrices K and K0，
K and K0The LogDet divergence is: dld(K,K0)＝tr(KK0 -1)-log det(KK0 -1) N, where n is the dimension of the input space, such that
Figure GDA0003150454250000062
After adding the relaxation variable γ, the expression is:
Figure GDA0003150454250000063
where ζ is the variable ζijColumn vector of (2), Xi,XjThe matrix is positively determined for the Riemann manifold.
Further, in step 3, the initial kernel matrix is defined according to the matrix A of the Mahalanobis distance
Figure GDA0003150454250000064
Is Xi,XjThe column vector of (1) is learned by adopting a Bregman iterative algorithm to obtain an optimal kernel matrix K*Approximation K0And enabling the finally obtained core matrix to meet the condition that the distance on the core space is equal to the geodetic distance on the Riemannian manifold, thereby achieving the purpose of keeping the geodetic distance. Connection between learning and nuclear learning based on measure
Figure GDA0003150454250000065
The optimal solution A suitable for the problem can be obtained according to measure learning*Then directly obtain K*. The invention adopts the method of substituting the obtained optimal Marfan matrix into linear transformation
Figure GDA0003150454250000066
And under the LogDet divergence, performing kernel learning to obtain an optimal kernel matrix, so that the distance in the kernel space is consistent with the geodesic distance in the Riemannian manifold.
For n × n matrices K and K0K and K0The LogDet divergence is:
Dld(K,K0)＝tr(KK0 -1)-logdet(KK0 -1) -n, where n is the dimension of the input space.
To is coming toThere is no feasible solution to prevent minimizing LogDet divergence, and a relaxation variable γ is added to the presentation to ensure the existence of a feasible solution. The constraint condition of the invention is that the distance between two points on the nuclear space is equal to the geodesic distance between two points on the Riemannian manifold, i.e. the order
Figure GDA0003150454250000067
After adding the relaxation variable γ, the expression is:
Figure GDA0003150454250000068
Figure GDA0003150454250000069
where ζ is the variable ζijThe column vector of (2).
Further, according to the obtained optimal kernel matrix K*And carrying out data classification.
Data classification description: by adopting linear classification, the similarity between two points can be directly calculated and then classified; the classification by the kernel method is to project data to a kernel space and then classify the data, and the classification is a nonlinear classification method. In this process, a kernel matrix needs to be calculated (generally, a kernel function such as a gaussian kernel function is directly adopted, or a kernel function or a kernel matrix is learned, which belongs to the second category here).
The invention also provides a Riemannian manifold preserving kernel learning device based on geodesic distance, which comprises: a memory, a processor, and a computer program stored in the memory and executable on the processor, the processor executing the computer program to operate in the units of:
the Riemann manifold calculation unit is used for calculating a characteristic covariance matrix of each sample data according to the data sample set and the covariance descriptor;
the Riemann geodesic distance calculation unit is used for calculating the Riemann geodesic distance according to the characteristic covariance matrix;
the iteration kernel optimization unit is used for learning the initial kernel matrix defined by the Riemann geodesic distance by adopting a Bregman iteration algorithm to obtain an optimal kernel matrix;
and the optimal kernel classification unit is used for carrying out data classification according to the obtained optimal kernel matrix.
The beneficial effect of this disclosure does: the disclosure provides a new nuclear learning method based on geodesic distance maintenance on Riemann manifold data of a digital image, which solves two problems of the nuclear learning method: 1) measure retention problem. According to the Log-Euclidean geodesic distance on the Riemannian manifold, a logarithmically symmetric positive definite matrix is processed into a column vector, and the distance of the symmetric positive definite matrix on the Riemannian manifold transformed to the Euclidean space is equal to the geodesic distance on the Riemannian manifold in a parameterized Mahalanobis distance learning mode. And meanwhile, expressing the Mahalanobis distance learning problem as the minimized LogDet divergence, and solving the optimal solution of the Mahalanobis distance matrix according to the Bregman optimization algorithm. 2) The problem of the fixed core method. The method finds the core which is suitable for the optimal problem in a core learning mode. While expressing the nuclear learning problem as minimizing LogDet divergence. And giving the initial value to the kernel matrix according to the linear transformation relation between the Mahalanobis distance matrix and the kernel matrix. And then performing kernel learning according to a Bregman optimization algorithm to obtain an optimal kernel matrix, so that the distance on the kernel space is consistent with the geodesic distance. Finally, the problems of Riemannian manifold measurement maintenance and single-core limitation are effectively solved.
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The foregoing and other features of the present disclosure will become more apparent from the detailed description of the embodiments shown in conjunction with the drawings in which like reference characters designate the same or similar elements throughout the several views, and it is apparent that the drawings in the following description are merely some examples of the present disclosure and that other drawings may be derived therefrom by those skilled in the art without the benefit of any inventive faculty, and in which:
FIG. 1 is a flow chart of a Riemannian manifold preservation kernel learning method based on geodesic distance according to the present disclosure;
fig. 2 is a diagram of a riemann manifold preserving kernel learning device based on geodesic distance according to the present disclosure.
Detailed Description
The conception, specific structure and technical effects of the present disclosure will be clearly and completely described below in conjunction with the embodiments and the accompanying drawings to fully understand the objects, aspects and effects of the present disclosure. It should be noted that the embodiments and features of the embodiments in the present application may be combined with each other without conflict.
Fig. 1 is a flowchart of a riemann manifold maintenance core learning method based on geodesic distance according to the present disclosure, and the following describes a riemann manifold maintenance core learning method based on geodesic distance according to an embodiment of the present disclosure with reference to fig. 1.
The invention mainly provides a Riemannian manifold maintenance kernel learning method based on geodesic distance, which has the following specific implementation modes:
let X be { X ═ X in a given dataset1…xnIn which xiIndicating a sample of data, i 1, …, n, the sample data of note is a matrix and does not need to be converted to a vector.
The feature covariance matrix for each sample is calculated. Here, a regional covariance operator method is used.
Let I be the sample picture of one-dimensional or three-dimensional color intensity. F is a W × H × d dimensional feature image extracted from I:
f (x, y) ═ phi (I, x, y), where W, H, d is the dimension of space,
the function phi is an arbitrary mapping such as image intensity, color, gradient, filter response, etc.
General feature vector F (x, y):
Figure GDA0003150454250000081
(x, y) denotes a position of the pixel, R (x, y), G (x, y), B (x, y) denotes a color value of the pixel,
Figure GDA0003150454250000082
respectively, the norm of the first differential with respect to x, y,
Figure GDA0003150454250000083
representing the norm of the second differential with respect to x, y, respectively. For the texture recognition task, a feature vector F (x, y) is typically constructed:
Figure GDA0003150454250000091
i (x, y) represents image intensity, as otherwise described above. In constructing the feature vector, the feature value is usually the norm of the first and second differential of the image intensity value with respect to x and y.
For a given rectangular area
Figure GDA0003150454250000092
{zk}k＝1…nIs a d-dimensional feature in region R. This region R is represented by a covariance matrix of d × d of the features in the region R:
Figure GDA0003150454250000093
u is the mean of the feature points. For the (i, j) th element of the covariance matrix in the equation:
Figure GDA0003150454250000094
spread the mean, the above equation may be changed to:
Figure GDA0003150454250000095
calculating the sum of the features of each dimension
Figure GDA0003150454250000096
And any two characteristicsSum of the products of features Z (i) Z (j)i,j＝1…nThe regional covariance can be obtained.
And learning the Mahalanobis distance under the condition of keeping the geodesic distance according to the calculated characteristic covariance matrix.
For any two symmetric positive definite matrixes Xi,XjPerforming a logarithmic operation to obtain Zi＝log(Xi),Zj＝log(Xj). After logarithmic calculation, Zi,ZjStill a matrix. Further, handle Zi,ZjThe matrix is processed into a column vector
Figure GDA0003150454250000097
Newly obtained column vector
Figure GDA0003150454250000098
Can be regarded as a point in Euclidean space, and by using a parameterized Mahalanobis distance form, can obtain:
Figure GDA0003150454250000099
learning of the above matrix a can be solved with LogDet divergence. I.e. where A and A are for n matrices0A and A0The LogDet divergence is:
Dld(A,A0)＝tr(AA0 -1)-log det(AA0 -1) -n, where n is the dimension of the input space.
There is no feasible solution to prevent minimizing LogDet divergence, and a relaxation variable γ is added to the representation to ensure an optimal solution. In the invention, in order to achieve the purpose of keeping geodesic distance, the point of the transformed symmetric positive definite matrix is obtained after learning the mahalanobis distance on the Euclidean space
Figure GDA0003150454250000101
After adding the relaxation variable γ, the learning problem for the matrix a can be expressed as:
Figure GDA0003150454250000102
Figure GDA0003150454250000103
where ζ is the variable ζijThe column vector of (2).
The process of optimizing the matrix A is an iterative process, and the steps are as follows
Figure GDA0003150454250000104
Figure GDA0003150454250000111
According to the optimal A, an initial kernel matrix can be obtained, and the initial kernel matrix can be defined
Figure GDA0003150454250000112
With the initial kernel matrix, learning the kernel matrix by adopting a Bregman iterative algorithm to obtain an optimal kernel matrix K*Approximation K0And the finally obtained nuclear matrix meets the condition that the distance on the nuclear space is equal to the geodetic distance on the Riemann manifold, so that the aim of keeping the geodetic distance is fulfilled.
The Bregman iterative algorithm generally iterates:
j: x- > R, convex function, non-negative. u ∈ X.
H: x- > R, convex function, non-negative. u ∈ X, f are known constants (image data, either matrix or vector).
X: the scope is either a convex set or a closed set.
Note that: the functional may have different specific expressions according to different specific problems. J. H, X are data items.
Firstly, initializing relevant parameters to be zero;
then theAnd then the formula u is iterated. Up to ukThe convergence condition is satisfied.
u is the Bregman distance of functional J.
p is the gradient of functional H.
The Bregman iterative algorithm can converge to a true optimal solution through multiple iterations.
For n × n matrices K and K0K and K0The LogDet divergence is:
Dld(K,K0)＝tr(KK0 -1)-log det(KK0 -1)-n，
where n is the dimension of the input space.
There is no feasible solution to prevent minimizing LogDet divergence, and a relaxation variable γ is added to the presentation to ensure the existence of a feasible solution. The constraint condition of the invention is that the distance between two points on the nuclear space is equal to the geodesic distance between two points on the Riemannian manifold, i.e. the order
Figure GDA0003150454250000121
After adding the relaxation variable γ, the expression is:
Figure GDA0003150454250000122
Figure GDA0003150454250000123
where ζ is the variable ζijThe column vector of (2).
Optimal solution K*The solution process of (2) is also an iterative process, as follows:
Figure GDA0003150454250000124
Figure GDA0003150454250000131
connection between learning and nuclear learning based on measure
Figure GDA0003150454250000132
The optimal solution A suitable for the problem can be obtained according to measure learning*Then directly obtain K*. The invention adopts the method of substituting the obtained optimal Marfan matrix into linear transformation
Figure GDA0003150454250000133
And under the LogDet divergence, performing kernel learning to obtain an optimal kernel matrix, so that the distance in the kernel space is consistent with the geodesic distance in the Riemannian manifold.
Further, according to the obtained optimal kernel matrix K*And carrying out data classification.
Data classification description: by adopting linear classification, the similarity between two points can be directly calculated and then classified; the classification by the kernel method is to project data to a kernel space and then classify the data, and the classification is a nonlinear classification method. In this process, a kernel matrix needs to be calculated (generally, a kernel function such as a gaussian kernel function is directly adopted, or a kernel function or a kernel matrix is learned, which belongs to the second category here).
A riemann manifold preservation nuclear learning device based on geodesic distance according to an embodiment of the present disclosure, as shown in fig. 2, is a riemann manifold preservation nuclear learning device based on geodesic distance according to the present disclosure, and includes: a processor, a memory, and a computer program, such as a read-write program, stored in the memory and executable on the processor. The processor implements the steps in the above-described respective read-write method embodiments when executing the computer program.
The device comprises: a memory, a processor, and a computer program stored in the memory and executable on the processor, the processor executing the computer program to operate in the units of:
the Riemann manifold calculation unit is used for calculating a characteristic covariance matrix of each sample data according to the data sample set and the covariance descriptor;
the Riemann geodesic distance calculation unit is used for calculating the Riemann geodesic distance according to the characteristic covariance matrix;
the iteration kernel optimization unit is used for learning the initial kernel matrix defined by the Riemann geodesic distance by adopting a Bregman iteration algorithm to obtain an optimal kernel matrix;
and the optimal kernel classification unit is used for carrying out data classification according to the obtained optimal kernel matrix.
The Riemannian manifold preservation kernel learning device based on the geodesic distance can be operated in computing equipment such as desktop computers, notebooks, palm computers, cloud servers and the like. The riemann manifold preservation core learning device based on geodesic distance can be operated by a device comprising, but not limited to, a processor and a memory. Those skilled in the art will appreciate that the example is merely an example of a riemann manifold maintenance core learning apparatus based on geodesic distance, and does not constitute a limitation of a riemann manifold maintenance core learning apparatus based on geodesic distance, and may include more or less than a proportion of components, or combine certain components, or different components, for example, the riemann manifold maintenance core learning apparatus based on geodesic distance may further include input and output devices, network access devices, buses, and the like.
The Processor may be a Central Processing Unit (CPU), other general purpose Processor, a Digital Signal Processor (DSP), an Application Specific Integrated Circuit (ASIC), an off-the-shelf Programmable Gate Array (FPGA) or other Programmable logic device, discrete Gate or transistor logic, discrete hardware components, etc. The general purpose processor may be a microprocessor or the processor may be any conventional processor or the like, the processor being the control center of the device for operating the riemann manifold maintenance nuclear learning device based on geodetic distances, and various interfaces and lines connecting various parts of the entire device for operating the riemann manifold maintenance nuclear learning device based on geodetic distances.
The memory may be configured to store the computer program and/or module, and the processor may implement the various functions of the riemann manifold preservation core learning device based on geodesic distance by executing or executing the computer program and/or module stored in the memory and calling data stored in the memory. The memory may mainly include a storage program area and a storage data area, wherein the storage program area may store an operating system, an application program required by at least one function (such as a sound playing function, an image playing function, etc.), and the like; the storage data area may store data (such as audio data, a phonebook, etc.) created according to the use of the cellular phone, and the like. In addition, the memory may include high speed random access memory, and may also include non-volatile memory, such as a hard disk, a memory, a plug-in hard disk, a Smart Media Card (SMC), a Secure Digital (SD) Card, a Flash memory Card (Flash Card), at least one magnetic disk storage device, a Flash memory device, or other volatile solid state storage device.
While the present disclosure has been described in considerable detail and with particular reference to a few illustrative embodiments thereof, it is not intended to be limited to any such details or embodiments or any particular embodiments, but it is to be construed as effectively covering the intended scope of the disclosure by providing a broad, potential interpretation of such claims in view of the prior art with reference to the appended claims. Furthermore, the foregoing describes the disclosure in terms of embodiments foreseen by the inventor for which an enabling description was available, notwithstanding that insubstantial modifications of the disclosure, not presently foreseen, may nonetheless represent equivalent modifications thereto.

Claims (4)

1. A riemann manifold preservation kernel learning method based on geodesic distance, the method comprising the steps of:
step 1, calculating a characteristic covariance matrix of each sample data according to a data sample set and a covariance descriptor;
step 2, calculating the Riemann geodesic distance according to the characteristic covariance matrix;
step 3, learning an initial kernel matrix defined by the Riemann geodesic distance by adopting a Bregman iterative algorithm to obtain an optimal kernel matrix;
step 4, data classification is carried out according to the obtained optimal kernel matrix;
in step 1, the method for calculating the feature covariance matrix of each sample data includes:
calculating a characteristic covariance matrix of each sample by adopting a region covariance operator method, setting I as a sample picture of three-dimensional color intensity, and F as a characteristic image of W multiplied by H multiplied by d dimensions extracted from I: f (x, y) ═ Φ (I, x, y), if I is a three-dimensional picture, F is a feature image of a sub-image taken from I, the size of F is in dimensions W × H × d, x, y is within the size range W, H, d is the number of elements in F (x, y), the function Φ is one of the mappings of image intensity, color, gradient, filter response, general feature vector F (x, y):
Figure FDA0003153306560000011
(x, y) denotes a position of the pixel, R (x, y), G (x, y), B (x, y) denotes a color value of the pixel,
Figure FDA0003153306560000012
respectively, the norm of the first differential of x, y,
Figure FDA0003153306560000013
the norm representing the second order differential of x, y, respectively, for the texture recognition task, the feature vector F (x, y) is constructed:
Figure FDA0003153306560000014
where I (x, y) represents image intensity, and commonly used eigenvalues in constructing the eigenvectors are the image intensity values, the norm of the first and second differentials with respect to x, y, for a given rectangular region
Figure FDA0003153306560000015
{zkD-dimensional features in the region R, k is 1 … n, n is a positive integer, and the region R is represented by a covariance matrix of d × d of features in the region R:
Figure FDA0003153306560000021
u is the mean of the feature points, expressed for the (i, j) th element of the covariance matrix in the formula,
Figure FDA0003153306560000022
and expanding the average value, changing the formula into,
Figure FDA0003153306560000023
calculating the sum of the features of each dimension
Figure FDA0003153306560000024
And the sum of the products of any two features z (i) z (j), i 1 … n, j 1 … n, n being a positive integer, to obtain the regional covariance CR；
In step 2, the method for calculating the riemann geodesic distance according to the feature covariance matrix comprises the following steps:
let X be { X ═ X in a given dataset1…xnIn which xiRepresenting data samples, i ═ 1, …, n, n is a positive integer;
for any two symmetric positive definite matrixes Xi,XjPerforming a logarithmic operation to obtain Zi＝log(Xi),Zj＝log(Xj) After logarithmic calculation, Zi,ZjStill a matrix, and further, Zi,ZjThe matrix is processed into a column vector
Figure FDA0003153306560000025
Newly obtained column vector
Figure FDA0003153306560000026
As EuropePoints in formula space are obtained using a parameterized mahalanobis distance form:
Figure FDA0003153306560000027
learning of the above formula matrix A is solved with LogDet divergence, i.e., where for n matrices A and A0A and A0The LogDet divergence is: dld(A,A0)＝tr(AA0 -1)-logdet(AA0 -1) N, in order to prevent no feasible solution for minimizing LogDet divergence, a relaxation variable gamma is added in the expression form to ensure that an optimal solution is obtained, and a point after transformation of the symmetric positive definite matrix is subjected to mahalanobis distance learning on Euclidean space to obtain
Figure FDA0003153306560000028
After adding the relaxation variable γ, the learning problem for the matrix a is expressed as:
Figure FDA0003153306560000029
Figure FDA00031533065600000210
where ζ is the variable ζijThe column vector of (a) is,
Figure FDA0003153306560000031
is the definition of the Mahalanobis distance between vectors of two points in Euclidean space, dAConverting the manifold data to Euclidean space for the geodetic distance between the data on the Riemann manifold, which is approximately equal to the geodetic distance by the Mahalanobis distance, i.e. the distance between the data on the Riemann manifold
Figure FDA0003153306560000032
To ensure that dgesFor geodetic distance, the matrix a contains distance information for ranging.
2. The Riemannian manifold guarantee based on geodesic distance of claim 1The method for learning by kernel is characterized in that in step 3, an initial kernel matrix is defined according to a matrix A of Mahalanobis distance
Figure FDA0003153306560000033
3. The Riemannian manifold maintenance kernel learning method based on geodesic distance as claimed in claim 1, wherein in step 3, a kernel matrix is learned by using Bregman iterative algorithm, so that the obtained optimal kernel matrix K*Approximation K0The resulting kernel matrix satisfies that the distance in kernel space is equal to the geodesic distance in Riemann manifold, for n × n matrices K and K0，
K and K0The LogDet divergence is: dld(K,K0)＝tr(KK0 -1)-logdet(KK0 -1) N, where n is the dimension of the input space, such that
Figure FDA0003153306560000034
After adding the relaxation variable γ, the expression is:
Figure FDA0003153306560000035
Xi,Xjpositively determining a matrix for the Riemann manifold, where ζ is a variable ζijThe column vector of (2).
4. A riemann manifold preserving kernel learning device based on geodesic distance, the device comprising: a memory, a processor and a computer program stored in the memory and executable on the processor, the processor implementing the steps of the method of riemann manifold preservation core learning based on geodesic distance of any one of claims 1 to 3 when executing the computer program.
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